Questions concerning smallest fraction between two given fractions.

Can we generalise $n$ for all possible fraction ranges, and if so, will $n$ always be of a certain form compared to $a,b,c,$ and $d$? (where $\frac{a}{b}<\frac{m}{n}<\frac{c}{d}$).

This answer uses your method.

In the following, $a,b,c,d,m,n$ are positive integers.

$\frac ab\lt \frac mn\lt \frac cd$ is equivalent to $$na\lt mb\qquad\text{and}\qquad \frac{md}{c}\lt n$$

From the second inequality, we can set $n=\lfloor\frac{md}{c}\rfloor+1$ where $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$.

So, from the first inequality, we get $$\left(\left\lfloor\frac{md}{c}\right\rfloor+1\right)a\lt mb\tag1$$

As a result, we can say that the smallest positive integer $n$ such that there exists an integer $m$ satisfying $\frac ab\lt\frac mn\lt\frac cd$ is given by$$\left\lfloor\frac{Md}{c}\right\rfloor+1$$ where $M$ is the smallest integer $m$ satisfying $(1)$.

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If $c=1$, then the smallest positive integer $n$ such that there exists an integer $m$ satisfying $\frac ab\lt\frac mn\lt\frac 1d$ is given by$$\left(\left\lfloor\frac{a}{b-da}\right\rfloor+1\right)d+1$$